Formal languages
and automata theory
Li Fan
Pumping Lemma
• Let L be a regular set. Then there is a
constant n such that if z is any word in
L, and |z|>=n, we may write z=uvw in
such a way that |uv|<=n, |v|>=1, and
for all i>=0, u(v^i)w is in L.
Furthermore, n is no greater than the
number of states of the smallest FA
accepting L.
Exercise
• The set L={0^(i^2) | i is an integer,
i>=1}, which consists of all strings of
0’s whose length is a perfect square,
is not regular.
• Prove that L = {(a^n)(b^k): n > k and
n>=0} is not regular.
Which of the following
languages are regular sets?
• {0^(2n) | n>=1}
• {(0^m)(1^n)(0^(m+n)) | m>=1 and
n>=1}
• {0^n | n is a prime}
• The set of all strings that do not have
three consecutive 0’s
• The set of all strings with an equal
number of 0’s and 1’s
• {x | x in (0+1)*, and x=x^R} x^R is x
written backward; for example,
(011)^R=110.
• {xwx^R | x,w in (0+1)+}.
Thank you